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Theorem mulcomnq 8763
Description: Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulcomnq  |-  ( A  .Q  B )  =  ( B  .Q  A
)

Proof of Theorem mulcomnq
StepHypRef Expression
1 mulcompq 8762 . . . 4  |-  ( A 
.pQ  B )  =  ( B  .pQ  A
)
21fveq2i 5671 . . 3  |-  ( /Q
`  ( A  .pQ  B ) )  =  ( /Q `  ( B 
.pQ  A ) )
3 mulpqnq 8751 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
4 mulpqnq 8751 . . . 4  |-  ( ( B  e.  Q.  /\  A  e.  Q. )  ->  ( B  .Q  A
)  =  ( /Q
`  ( B  .pQ  A ) ) )
54ancoms 440 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  .Q  A
)  =  ( /Q
`  ( B  .pQ  A ) ) )
62, 3, 53eqtr4a 2445 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( B  .Q  A ) )
7 mulnqf 8759 . . . 4  |-  .Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5536 . . 3  |-  dom  .Q  =  ( Q.  X.  Q. )
98ndmovcom 6173 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( B  .Q  A ) )
106, 9pm2.61i 158 1  |-  ( A  .Q  B )  =  ( B  .Q  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717    X. cxp 4816   ` cfv 5394  (class class class)co 6020    .pQ cmpq 8657   Q.cnq 8660   /Qcerq 8662    .Q cmq 8664
This theorem is referenced by:  recmulnq  8774  recrecnq  8777  halfnq  8786  ltrnq  8789  addclprlem1  8826  addclprlem2  8827  mulclprlem  8829  mulclpr  8830  mulcompr  8833  distrlem4pr  8836  1idpr  8839  prlem934  8843  prlem936  8857  reclem3pr  8859  reclem4pr  8860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6841  df-ni 8682  df-mi 8684  df-lti 8685  df-mpq 8719  df-enq 8721  df-nq 8722  df-erq 8723  df-mq 8725  df-1nq 8726
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